Definition:Axiom/Formal Systems/Axiom Schema

Definition

Let $\LL$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\LL$ is to specify its axiom schemata.

An axiom schema is a well-formed formula $\phi$ of $\LL$, except for it containing one or more variables which are outside $\LL$ itself.

This formula can then be used to represent an infinite number of individual axioms in one statement.

Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs.

Each WFF $\psi$ that results from $\phi$ by a valid choice of values for all the variables is then an axiom of $\mathscr P$.

Also known as

An axiom schema is also known as just a schema.

Also see

• ZFC requires Axiom Schema
• Peano Arithmetic requires Axiom Schema

• Results about axiom schemata can be found here.

Linguistic Note

The plural of axiom schema is correctly axiom schemata, but it is commonplace to see the word schemas used for schemata.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): schema (plural schemata)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): schema (plural schemata)